To determine the equation that relates $x$ and $y$ from the given table, we need to examine the points $(-1, \_), (0, \_), (1, 5), (2, 3)$ and identify a potential pattern.

Let's solve:

Step 1: Analyze the data points

1. At $x = 1$, $y = 5$.
2. At $x = 2$, $y = 3$.

sympy as sp

Define the variables

x = sp.Symbol('x') y = sp.Function('y')

Create the known points

points = [(-1, sp.Symbol('y(-1)')), (0, sp.Symbol('

It seems there was an error in my initial code structure. Let me correctly process the given points to find the equation. Given points from the table points = [(-1, None), (0, None), (1, 5), (2, 3)]

Calculate the slope between known points (1, 5) and (2, 3)

slope = (3 - 5) / (2 - 1) # (y2 - y1) / (x2 - x1)

Use point-slope form to find the equation of the line

y - y1 = m(x - x1), using point (1, 5)

m = slope x = sp.Symbol('x') equation = m * (x - 1) + 5 # Substitute slope and point into the equation

equationIt seems the earlier context was reset; let me re-import the required tools and resolve this step by step. Re-importing sympy import sympy as sp

Define the known points

x = sp.Symbol('x') # Variable x points = [(1, 5), (2, 3)]

Calculate the slope between the points (1, 5) and (2, 3)

slope = (3 - 5) / (2 - 1) # (y2 - y1) / (x2 - x1)

Use point-slope form: y - y1 = m(x - x1), using the point (1, 5)

m = slope equation = m * (x - 1) + 5 # Substitute slope and point

equation.simplify() # Simplify the equation to standard formThe equation of the line relating $x$ and $y$ based on the given points is:

$y = 7 - 2x$

Steps Summary:

1. The slope ($m$) between the points $(1, 5)$ and $(2, 3)$ was calculated as: $m = \frac{3 - 5}{2 - 1} = -2$
2. Using the point-slope formula, $y - y_1 = m(x - x_1)$, with point $(1, 5)$, the equation becomes: $y - 5 = -2(x - 1)$
3. Simplifying: $y = 7 - 2x$

Would you like me to:

1. Explain how to verify this equation with the other data points?
2. Determine the $y$-values for $x = -1$ and $x = 0$?
3. Explore whether this is the only possible relationship?
4. Explain the derivation of the point-slope formula in detail?
5. Extend this to non-linear equations if required?

Tip:

Always verify an equation by substituting all given data points to ensure consistency!